ALGORITHMS FOR SELF-CONSISTENT SIMULATIONS OF BEAM-INDUCED PLASMA IN MUON COOLING DEVICES

A code called SPACE for the simulation of beam- induced plasma in gas-filled RF cavities has been devel- oped. The core code uses the particle-in-cell (PIC) method for the Maxwell equations coupled to the dynamics of particles. In the SPACE code, the electromagnetic PIC methods are coupled with probabilistic treatment of atomic physics processes. The aim of the project is to demonstrate the feasibility of the RF cavity with muon collider beams. The most critical part is investigating how intense muon beams influence the plasma dynamics. Initial phase of sim- ulations demonstrating the plasma generation and its short time dynamics in the high pressure hydrogen filled RF test cell has been performed. space and time. Electric charges are represented by dis- crete macroparticles coupled with electromagnetic fields by the action of Lorentz forces and electric currents. When one solves Maxwells equations analytically, solving two equations describing the Faraday and Ampere laws is suf- ficient as the equations expressing the Gauss law and the divergence-free constraint for the magnetic field are invari- ants of motion. The numerical method of Yee (2) for elec- tromagnetic fields in vacuum also preserves the divergence- free properties of the electric and magnetic fields. To deal with the last two equations in the presence of charges and currents, the rigorous charge conservation method was de- veloped within the PIC framework in (3). The method calculates electric currents along computational grid edges by solving a geometrical problem of sweeping the com- putational mesh by finite volumes associated with each macroparticle. By using this method, we find first a set of initial conditions consistent with the Maxwell equations by either solving the Poisson problem for the electric potential or by a superposition of electric fields and changes created by each particle. Then only the first two Maxwell equa- tions and the Newton-Lorentz equation are solved numer- ically thus avoiding solving the Poisson problem at each time step. A schematic of processes computed at each time step is depicted in Figure 1. We would like to note that the conser- vative Leapfrog time discretization scheme for the Newton- Lorentz equations of particle motion becomes implicit. We use the Boris scheme for the time update (4), which is a modification of the Leapfrog scheme resulting in an ex- plicit and conservative scheme. We also implement mod- ifications of the Boris scheme proposed in (5) for dealing with rapidly accelerating particles for which the relativistic factor is not constant. For accurate simulations of electromagnetic fields in ge- ometrically complex structures, we have implemented the embedded boundary method (6) in a stand-alone Maxwell equation solver. The coupling of algorithms for complex boundaries with the main electromagnetic PIC code will be performed in the next phase.